I would always write them without a space, but that's just my habit, and shouldn't be taken as a definitive answer. I don't think it's a bad habit to use these symbols outside of a paper on logic. I think most mathematicians would know what they mean. However, if you think it is clearer to write 'for all $x$' instead of $\forall x$ and it doesn't get in the way of your exposition, then feel free to spell it out in longhand instead.
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Chris TaylorSep 27 '11 at 23:33

@Chris: Thanks for the comment. After seeing the answers I have decided to leave my paper unchanged, since the way I typed it seems OK to others.
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B MSep 27 '11 at 23:46

And for a contrasting view, I find it somewhat jarring when one mixes symbols and English text. One can embed a symbolic formula in English text, of course, but not the other way around except in informal or very special situations. So if you use a symbol for the quantifier, you should let the entire scope of the quantified variable be a symbolic expression. Otherwise it just looks like textspeak.
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Henning MakholmSep 27 '11 at 23:58

I think that spelling out "for all" and "there exists" is the thing to do if it appears as part of an English sentence. As part of a displayed formula, $\forall$ and $\exists$ are fine, but I'd still prefer spelling it out unless there is a specific reason for doing otherwise.
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George LowtherSep 28 '11 at 0:07

3, yes. Use these symbols only when doing symbolic logic, or when doing quick abbreviations for your own use. Ordinarily (especially in papers) write things out in words.
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GEdgarSep 28 '11 at 0:17

2 Answers
2

My view is that you can leave it to $\TeX$ to sort out the spacing, trusting Donald Knuth, so the first of each of your two examples.

I would say that it is acceptable to use in mathematics (which I do not see as a subset of logic) as in $$\forall n \in \mathbb{N}: \sum_{i=0}^n i = \frac{n(n+1)}{2}$$ but rare outside logic and mathematics.

Thanks for the answer, Henry. By outside of logic, I did imply the universal set of Mathematics. I was mainly concerned when there was more than one quantity involved since TEX does typeset a space after the comma.
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B MSep 27 '11 at 23:44

Thanks for the answer, mixed math. I was about to revise my entire paper, but it seems the general consensus is to prefer not having a space, which is the way I typed it, so I will leave it as is.
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B MSep 27 '11 at 23:45

@BM I would be curious to see your paper. Have you uploaded a preprint or anything?
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mixedmath♦Sep 27 '11 at 23:50